Basic invariants
| Dimension: | $12$ |
| Group: | $S_3 \wr C_3 $ |
| Conductor: | \(212299175847866409\)\(\medspace = 3^{26} \cdot 17^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.5.6586148313.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 18T206 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_3\wr C_3$ |
| Projective stem field: | Galois closure of 9.5.6586148313.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - 3x^{6} - 9x^{5} + 9x^{4} + 9x^{3} - 9x^{2} + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$:
\( x^{3} + x + 40 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 35 a^{2} + 26 a + 22 + \left(10 a^{2} + 35 a + 27\right)\cdot 43 + \left(36 a^{2} + 9 a + 33\right)\cdot 43^{2} + \left(26 a^{2} + 4\right)\cdot 43^{3} + \left(33 a^{2} + 17 a + 42\right)\cdot 43^{4} + \left(a^{2} + 39 a + 5\right)\cdot 43^{5} + \left(12 a^{2} + 37 a + 6\right)\cdot 43^{6} + \left(2 a^{2} + 17 a + 7\right)\cdot 43^{7} + \left(19 a^{2} + 40 a + 34\right)\cdot 43^{8} + \left(23 a^{2} + 18 a + 10\right)\cdot 43^{9} +O(43^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 27 a^{2} + 10 a + 4 + \left(30 a^{2} + 32 a + 31\right)\cdot 43 + \left(29 a^{2} + 18 a + 18\right)\cdot 43^{2} + \left(8 a^{2} + 12 a + 9\right)\cdot 43^{3} + \left(20 a^{2} + 2 a + 30\right)\cdot 43^{4} + \left(17 a^{2} + 4 a + 37\right)\cdot 43^{5} + \left(7 a^{2} + 10 a + 25\right)\cdot 43^{6} + \left(33 a^{2} + 32 a + 1\right)\cdot 43^{7} + \left(40 a^{2} + 6 a + 26\right)\cdot 43^{8} + \left(23 a^{2} + 23 a + 29\right)\cdot 43^{9} +O(43^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 24 a^{2} + 7 a + 17 + \left(a^{2} + 18 a + 27\right)\cdot 43 + \left(20 a^{2} + 14 a + 33\right)\cdot 43^{2} + \left(7 a^{2} + 30 a + 28\right)\cdot 43^{3} + \left(32 a^{2} + 23 a + 13\right)\cdot 43^{4} + \left(23 a^{2} + 42 a + 42\right)\cdot 43^{5} + \left(23 a^{2} + 37 a + 10\right)\cdot 43^{6} + \left(7 a^{2} + 35 a + 34\right)\cdot 43^{7} + \left(26 a^{2} + 38 a + 25\right)\cdot 43^{8} + \left(38 a^{2} + 2\right)\cdot 43^{9} +O(43^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 32 a^{2} + 15 a + 36 + \left(34 a^{2} + 30 a + 33\right)\cdot 43 + \left(33 a^{2} + 2 a + 35\right)\cdot 43^{2} + \left(17 a^{2} + 21 a + 29\right)\cdot 43^{3} + \left(16 a^{2} + 18 a + 27\right)\cdot 43^{4} + \left(9 a^{2} + 4 a + 3\right)\cdot 43^{5} + \left(38 a^{2} + 21 a + 32\right)\cdot 43^{6} + \left(12 a^{2} + 41 a + 16\right)\cdot 43^{7} + \left(33 a^{2} + 10 a + 35\right)\cdot 43^{8} + \left(4 a^{2} + 4 a + 16\right)\cdot 43^{9} +O(43^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 35 a^{2} + 39 a + 10 + \left(35 a^{2} + 6 a + 7\right)\cdot 43 + \left(42 a^{2} + 21 a + 20\right)\cdot 43^{2} + \left(25 a^{2} + 33 a + 12\right)\cdot 43^{3} + \left(10 a^{2} + 24 a + 42\right)\cdot 43^{4} + \left(16 a^{2} + a + 22\right)\cdot 43^{5} + \left(37 a + 38\right)\cdot 43^{6} + \left(24 a^{2} + 19 a + 30\right)\cdot 43^{7} + \left(41 a^{2} + 25 a + 21\right)\cdot 43^{8} + \left(38 a^{2} + 20 a + 31\right)\cdot 43^{9} +O(43^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 19 a^{2} + 32 a + 40 + \left(15 a^{2} + 5 a + 1\right)\cdot 43 + \left(9 a^{2} + 19 a + 30\right)\cdot 43^{2} + \left(42 a^{2} + 31 a\right)\cdot 43^{3} + \left(15 a^{2} + 42 a + 16\right)\cdot 43^{4} + \left(17 a^{2} + 36 a + 16\right)\cdot 43^{5} + \left(4 a^{2} + 27 a + 15\right)\cdot 43^{6} + \left(6 a^{2} + 24 a + 38\right)\cdot 43^{7} + \left(11 a^{2} + 6 a + 28\right)\cdot 43^{8} + \left(42 a^{2} + 18 a + 37\right)\cdot 43^{9} +O(43^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 27 a^{2} + 40 a + 19 + \left(5 a^{2} + 17 a + 1\right)\cdot 43 + \left(23 a^{2} + 7 a + 7\right)\cdot 43^{2} + \left(9 a^{2} + 22 a + 30\right)\cdot 43^{3} + \left(37 a + 6\right)\cdot 43^{4} + \left(3 a^{2} + 41 a + 14\right)\cdot 43^{5} + \left(19 a^{2} + 10 a + 22\right)\cdot 43^{6} + \left(11 a^{2} + 30 a + 22\right)\cdot 43^{7} + \left(18 a^{2} + 21 a + 20\right)\cdot 43^{8} + \left(8 a^{2} + 21 a + 25\right)\cdot 43^{9} +O(43^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 32 a^{2} + 28 a + 20 + \left(16 a^{2} + a + 31\right)\cdot 43 + \left(40 a^{2} + 14 a + 7\right)\cdot 43^{2} + \left(16 a^{2} + 11 a + 41\right)\cdot 43^{3} + \left(36 a^{2} + 26 a\right)\cdot 43^{4} + \left(23 a^{2} + 9 a + 35\right)\cdot 43^{5} + \left(26 a^{2} + 20 a + 15\right)\cdot 43^{6} + \left(34 a^{2} + 14\right)\cdot 43^{7} + \left(12 a^{2} + 39 a + 1\right)\cdot 43^{8} + \left(20 a^{2} + 5 a + 23\right)\cdot 43^{9} +O(43^{10})\)
|
| $r_{ 9 }$ | $=$ |
\( 27 a^{2} + 18 a + 4 + \left(20 a^{2} + 23 a + 10\right)\cdot 43 + \left(22 a^{2} + 21 a + 28\right)\cdot 43^{2} + \left(16 a^{2} + 9 a + 14\right)\cdot 43^{3} + \left(6 a^{2} + 22 a + 35\right)\cdot 43^{4} + \left(16 a^{2} + 34 a + 36\right)\cdot 43^{5} + \left(40 a^{2} + 11 a + 4\right)\cdot 43^{6} + \left(39 a^{2} + 12 a + 6\right)\cdot 43^{7} + \left(11 a^{2} + 25 a + 21\right)\cdot 43^{8} + \left(14 a^{2} + 15 a + 37\right)\cdot 43^{9} +O(43^{10})\)
|
Generators of the action on the roots $r_1, \ldots, r_{ 9 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 9 }$ | Character value |
| $1$ | $1$ | $()$ | $12$ |
| $9$ | $2$ | $(7,8)$ | $4$ |
| $27$ | $2$ | $(1,2)(4,5)(7,8)$ | $0$ |
| $27$ | $2$ | $(5,6)(7,8)$ | $0$ |
| $6$ | $3$ | $(1,2,3)$ | $0$ |
| $8$ | $3$ | $(1,2,3)(4,5,6)(7,8,9)$ | $3$ |
| $12$ | $3$ | $(1,2,3)(4,5,6)$ | $-3$ |
| $36$ | $3$ | $(1,4,7)(2,5,8)(3,6,9)$ | $0$ |
| $36$ | $3$ | $(1,7,4)(2,8,5)(3,9,6)$ | $0$ |
| $18$ | $6$ | $(1,2,3)(7,8)$ | $-2$ |
| $18$ | $6$ | $(4,5,6)(7,8)$ | $-2$ |
| $36$ | $6$ | $(1,2,3)(4,5,6)(7,8)$ | $1$ |
| $54$ | $6$ | $(1,2,3)(5,6)(7,8)$ | $0$ |
| $108$ | $6$ | $(1,4,7,2,5,8)(3,6,9)$ | $0$ |
| $108$ | $6$ | $(1,8,5,2,7,4)(3,9,6)$ | $0$ |
| $72$ | $9$ | $(1,5,8,2,6,9,3,4,7)$ | $0$ |
| $72$ | $9$ | $(1,8,6,3,7,5,2,9,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.